# Using Inequalities to Solve Equations

Suppose we need to solve equation f(x)=g(x) and suppose there exists such number A that is the maximum value of function y=f(x) and minimum value of function y=g(x).

Then roots of the equation f(x)=g(x) are common roots of the equations f(x)=A,g(x)=A and only they.

Example. Solve x^2+1/x^2=2-(x-1)^4.

If we directly begin solve this equation, then we will end up with the equation of 6-th degree that is not very pleasant to solve. So, we need some trick here.

Note, that by Cauchy's inequality (x^2+1/x^2)/2>=sqrt(x^2*1/x^2) or x^2+1/x^2>=2.

From another side, since (x-1)^4>=0 then 2-(x-1)^4<=2.

This means that roots of initial equation are common roots of equations x^2+1/x^2=2 and 2-(x-1)^4=2 (if there exist some roots).

From equation x^2+1/x^2=2 we obtain that x_1=1,x_2=-1.

Frrom equation 2-(x-1)^4=2 we obtain that x=1.

Common root of these two equations is x=1. Therefore, x=1 is the only root of the initial equation.